A transformation‐free HOC scheme for steady convection–diffusion on non‐uniform grids

TLDR: In this paper, a higher order compact finite difference solution procedure has been proposed for the steady two-dimensional convection-diffusion equation on non-uniform orthogonal Cartesian grids involving no transformation from the physical space to the computational space.

Abstract: A higher order compact (HOC) finite difference solution procedure has been proposed for the steady two-dimensional (2D) convection–diffusion equation on non-uniform orthogonal Cartesian grids involving no transformation from the physical space to the computational space. Effectiveness of the method is seen from the fact that for the first time, an HOC algorithm on non-uniform grid has been extended to the Navier–Stokes (N–S) equations. Apart from avoiding usual computational complexities associated with conventional transformation techniques, the method produces very accurate solutions for difficult test cases. Besides including the good features of ordinary HOC schemes, the method has the advantage of better scale resolution with smaller number of grid points, with resultant saving of memory and CPU time. Gain in time however may not be proportional to the decrease in the number of grid points as grid non-uniformity imparts asymmetry to some of the associated matrices which otherwise would have been symmetric. The solution procedure is also highly robust as it computes complex flows such as that in the lid-driven square cavity at high Reynolds numbers (Re), for which no HOC results have so far been seen. Copyright © 2004 John Wiley & Sons, Ltd.